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The Classification of the First Order Ordinary Differential Equations with the Painlev´eProperty The Classical and a Modern Algebro-Geometric Approach Version 1.0 Georg Muntingh Preface and Acknowledgments I would like to acknowledge my considerable debt to many people who helped me. First of all I want to thank my supervisors, Marius van der Put and Jaap Top. Their doors were always open, and they never seemed to tire of ex- plaining something to me for the second or even the third time, encouraging me to keep trying. Secondly my gratitude goes out to Professor Masahiko Saito from Kobe University for his talk in Utrecht that indirectly led to this thesis and his help with references for first order equations. Several people are responsible for making the text more readable and clear, both on the mathematical and on the linguistical part. For that my thanks go to Monique van Beek, Jeroen Sijsling, Laurens van der Starre and of course my girlfriend Annett. Moving to Oslo to live with her formed the primary motivation to finish up my thesis. Furthermore I am of course indebted to my family for supporting me all my life. I would like to thank several people who provided me with the facilities I needed to write this thesis, in particular the system operators Harm Paas, Jurjen Bokma and Peter Arendz for the GNU/Linux Debian system at the Department of Mathematics at the Rijksuniversiteit Groningen, the clean- ing lady Anja who was always cheerful in the morning and Ineke from the administration for the invigorating chats and the many cups of coffee. For typesetting I used LATEX 2ε and the very convenient LATEX editor Kile. The pictures were made with The Gimp, Gnuplot and Dia, and the frontispiece was inspired by the logo of Wikipedia. The first chapter will serve as an introduction to Painlev´eTheory, giving some motivation and intuitive definitions. At the end of the chapter several questions will be posed that will be discussed later in the thesis. The second chapter will deal with a large part of the mathematics that is needed later on, especially in the chapter on modern theory. In the third chapter an i overview of the historical development of Painlev´eTheory will be given, together with rewritten classical theorems and a rewritten classical proof. After that, in the fourth chapter, a detailed modern theory will be presented, followed by the fifth and final chapter containing conclusions and suggestions for future work. At the end of the document one can find an index of some terminology and names, referring to the page where they occurred first, and a list of symbols accompanied by a short description. ii CONTENTS 1 Introduction 2 1.1 Problematic Points . 2 1.2 The Painlev´eProperty . 4 1.3 First Order Equations with the PP . 6 2 Prerequisites 9 2.1 Fields of Functions . 9 2.2 Factorizing a Polynomial Differential Equation . 12 2.3 Local Rings and Valuations . 13 2.4 Ramification and Branch Points . 15 2.5 Differential Function Fields . 16 3 Classical Painlev´eTheory 19 3.1 First Order Painlev´eTheory in the Literature . 19 3.2 Classical Proof of the Theorem of Briot and Bouquet . 24 3.3 The Algorithm Indicated by the Theorem of Briot and Bouquet 31 3.4 Fuchs’s Criterion . 32 4 Modern Painlev´eTheory 34 4.1 The Setting and Theme . 34 4.2 The Algebraic Painlev´eProperty . 37 4.3 The Riccati Equation . 38 4.4 The Generalized Weierstrass Equation . 40 4.5 Classification of the First Order Autonomous Equations . 42 4.6 Classification of the First Order Equations . 46 5 Discussion 57 5.1 Conclusions . 57 iii 5.2 Future Work . 58 A Implementation of a Painlev´eTest in Maple 59 B The Painlev´eProperty in Physics 61 List of Symbols 67 iv LIST OF FIGURES 1 A portrait of Paul Painlev´efrom 1929 . 1 1.1 A picture of the Riemann surface corresponding to the loga- rithm, above a disk in the complex plane. 3 2.1 Several structures of functions together with their relations. 11 3.1 Two 19th century mathematicians who layed the foundations for Painlev´eTheory. 20 3.2 Two mathematicians who solved the general case for first or- der equations. 22 3.3 Two mathematicians who redid some classical Painlev´eTheory. 24 4.1 A schematic representation of the situation in the proof of Proposition 4.28. 50 B.1 A caricature of Paul Painlev´efrom 1932 . 69 v Figure 1: A portrait of Paul Painlev´efrom 1929 1 CHAPTER 1 Introduction Les Math´ematiques constituent un continent solidement agenc´e, dont tous les pays sont bien reli´esles uns aux autres; l’oeuvre de Paul Painlev´eest une ˆıle originale et splendide dans l’oc´ean voisin,– H. Poincar´e he class of all differential equations is enormous and very complicated T to study in general. The best one can do is to restrict our research to a class of differential equations that is easy enough to say sensible things about and wide enough to describe a wide spectrum of phenomena. This thesis is about such a class, namely the class of first order ordinary differential equations with the Painlev´eProperty. Before we define the Painlev´eProperty, we give in Section 1.1 some conceptual definitions of the problematic points of a differential equation. Armed with these notions, we give a conceptual definition of the Painlev´e Property in Section 1.2 and motivate why we should study equations with this property. Finally, in Section 1.3, we shall restrict ourselves to the case of first order ordinary differential equations and discuss what kind of questions we can ask ourselves. The remainder of the thesis will then be concerned with the answers to these questions. 1.1 Problematic Points In this section, we will look at some of the problems that can occur regarding the solutions of differential equations. To be more precise, we shall introduce three types of so-called problematic points. When we know which problems can occur in this context, we can restrict ourselves to differential equations that do not have these problems and try to examine this simplified case. This is exactly what is done in Painlev´eTheory. 2 Figure 1.1: A picture of the Riemann surface corresponding to the logarithm, above a disk in the complex plane. An example of a problematic point is a so called branch point. This is a point in which multivaluedness of the solution occurs. This means that there does not exist a neighborhood of the point in the complex plane in which we can define a solution, but there does exist a neighborhood in the universal covering space on which we can. Let us illustrate this with an example. Example 1.1 (Branch point of the logarithm). Consider the initial value prob- lem 1 f 0(z) = , f(1) = 0. z In a neighborhood of the point 1, this equation clearly has a solution f(z) := log z. From function theory, however, we know that we cannot get a power series solution in a neighborhood of 0. Furthermore, we cannot extend the solution in a neighborhood of 1 to a function that has a power series expansion in a punctured neighborhood of 0. So what can we do? The answer is that we can construct a larger space, called a Riemann surface, on which a solution can be defined. Maximal solutions that we were able to define on the complex plane, for instance the analytic function log : C\R≤0 → C that restricts to the ordinary logarithm on R>0, then appear as projections to C of restrictions to a certain sheet of our grand solution on the Riemann surface (see figure 1.1). In general we call a point a branch point of the differential equation if it has a punctured neighborhood U in which a power series solution of the differential equation can everywhere determined locally, yet there exists no 3 global solution on U restricting to a power series everywhere, no matter how small U is chosen. Another problematic point is a pole. This is a point for which the solution goes to infinity when we approach this point, but there exists a monomial zk such that the product of zk and the solution can locally be expressed as a convergent power series. Another way to say this is that the function has a pole in 0 if it can locally be expressed as a Laurent series in z. An easy example of a differential equation with a pole is the following. Example 1.2 (Pole). Consider the initial value problem 1 f 0(z) = − , f(1) = 1. z2 1 This initial value problem clearly has a unique solution f(z) := z , which goes to infinity at z = 0. If we multiply this solution by z1, then we obtain the function that equals 1 in every point. This function is clearly the convergent power series 1 + 0 · z + 0 · z2 + ··· . Therefore this initial value problem has a pole at z = 0. Another type of singularities is the type of essential singularities. These are much worse than poles. Example 1.3 (Essential singularity). Consider the initial value problem 1 f 0(z) = − f(z), f(1) = e. z2 This equation has a unique solution f(z) := e1/z, which has a singularity at z = 0. Since there does not exist a natural number k such that zke1/z is a convergent power series at z = 0, this is an essential singularity. We shall call all these three types of points problematic points.

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